capital asset pricing model & mutual fund performance ... · pdf fileheinz zimmermann...

Heinz Zimmermann – Asset Pricing and Performance Review Seite 1

Capital Asset Pricing Model & Mutual Fund Performance Studies –

Review and Evidence

Prof. Dr. Heinz Zimmermann and Elmar MertensWirtschaftswissenschaftliches Zentrum WWZ

Universität BaselVersion: 25 Januar, 2002


The Basic Paradigm: “Market Efficiency”

• Information affects prices• Prices “reflect” information• This makes only sense in the context of incomplete, or

heterogeneous information• So, information aggregation is the issue• Are competitive price systems able to aggregate information, all

information? What information?• Costs of information processing, the free-rider problem, and the

impossibility of informationally efficient markets (or better: a fullyrevealing price system) in equilibrium


Market Efficiency – The history

• The Martingale model: Bachelier 1900• The empirical record on Random Walks: Cowles, Kendall,

Cootner, Fama …• Random Walk and predictability: Samuelson 1965, 1972• The Fama definition 1970• Grossman/ Stiglitz 1976: Information aggregation and

competitive price systems• Grossman/ Stiglitz 1980, Ippolito 1989: Costly information, and

the impossibility of informationally efficient markets• Market efficiency and volatility bounds: Present value relations

and discouting


The CAPM

Beta

Expected Return

Riskfree Rate

Expected Risk Premium

� �RR MiMi ��


Who has developed the CAPM?

• Jack Treynor 1961 unpublished• William Sharpe 1964• John Lintner 1965• Ian Mossin 1966• Michael Jensen 1968


CAPM – Econometric issues

• How to estimate expected returns?• Estimated instead of true betas (Miller/ Scholes 1972)• Specification of market portfolio (Roll 1977, Stambaugh 1982)• Time variation of betas• Time variation of expected risk premia• Time horizon, no riskless asset (Black 1972)• Nominal or real returns?• Non-normality of returns


CAPM – Classic tests

The basic test methodologies:• Ex ante CAPM must be transformed to an ex post test equation• Tests for individual stocks vs. beta-grouped portfolios• Test strategy 1: Time-series regressions• Test strategy 2: Cross-sectional regression based on average

returns• Test strategy 3: Time-series of cross-sectional regressions


Black/ Jensen/ Scholes 1972

PF � Überrendite � �

[%]

1 1.56 2.13 -0.083 0.962 1.38 1.77 -0.194 0.993 1.25 1.71 -0.065 0.994 1.16 1.63 -0.017 0.995 1.06 1.45 -0.054 0.996 0.92 1.37 0.059 0.987 0.85 1.26 0.046 0.988 0.75 1.15 0.081 0.989 0.63 1.09 0.197 0.9610 0.49 0.91 0.201 0.90

Markt 1.0 1.42

� �R R R Rit ft i i M t ft it� � � � ��


Fama/McBeth 1973

Periode �0 [%] �1 �2 �3

1935-40 0.09 0.016 -0.003 0.0031941-45 0.15 0.007 0.001 0.1771946-50 0.11 0.014 -0.004 -0.0311951-55 0.23 0.028 -0.011 -0.0441956-60 1.03 -0.005 -0.002 0.0981961-68 -0.17 0.009 0.001 0.096

1935-68 0.20 0.011 -0.003 0.052

Rit t t i t i t ei it� � � � �� 0 1 22

3


CAPM – Extensions I

• No riskfree asset : Zero-beta CAPM. Black 1972• Non-traded assets: Mayers 1972• Intertemporal CAPM, Non-stationarities: Merton 1973• Dividends, taxes: Litzenberger, Scholes, …• Foreign exchange risk: Solnik 1974• Inflation and PPP risk: Friend/ Landskroner/Losq 1975, Sercu

1980, Adler Dumas 1983• Investment restrictions: Stulz 1983


CAPM – Extensions II

• Short-sale restrictions: Ross 1975• Heterogeneous expecations: Williams 1977• Consumption risk: Breeden 1979, Stulz 1981• LPMs instead of variances: Harlow/ Rao 1989


CAPM – Roll‘s 1977 critique

• There is a linear pricing relationship between the expectedreturns of any portfolios and their betas to any MV-frontierportfolio

• Linear pricing relationships do not tell us anything about theunderlying economic equilibrium! They hold by no-arbitragealone.

• If the market portfolio is not ex ante efficient, then therelationship between risk and expected return is not necessarilylinear.

• Any test of market efficiency is at most a test about the ex anteefficiency of the market portfolio.


CAPM – Anomalies

• P/E: Basu 1977• Size: Banz 1981• Dividend yields and the Dow Theory: Fama/French …• Value-growth: Capaul/ Rowley/ Sharpe• Seasonalities (January, day of the week, …): Keim, Reinganum,

etc.


Some Fama-French results

Durch-schnitt

tief-Beta

2 3 4 5 6 7 8 9 hoch-Beta

Durch-schnitt

1.3 1.3 1.3 1.4 1.3 1.3 1.3 1.2 1.2 1.3 1.1

klein 1.5 1.7 1.6 1.8 1.6 1.5 1.5 1.4 1.6 1.5 1.42 1.3 1.3 1.4 1.4 1.4 1.7 1.6 1.4 1.3 1.3 1.13 1.2 1.1 1.3 1.2 1.7 1.3 1.1 1.3 1.4 1.3 0.84 1.3 1.3 1.1 1.5 1.1 1.3 1.1 1.4 1.2 1.4 1.05 1.3 1.3 1.4 1.4 1.5 1.4 1.2 1.1 1.3 1.2 1.16 1.2 1.1 1.5 1.3 1.2 1.2 1.2 1.2 1.0 1.1 1.07 1.1 1.0 1.2 1.3 1.1 1.2 1.1 1.2 0.6 1.3 0.88 1.1 1.1 1.1 1.4 1.2 1.3 1.0 1.2 1.0 1.0 0.99 1.0 1.0 0.9 1.0 1.1 1.1 1.2 0.9 0.8 0.9 0.6gross 0.9 1.0 0.9 1.1 0.9 0.9 0.9 1.0 0.7 0.7 0.6

Datengrundlage: Zeitperiode 1963-90, monatliche Durchschnittsrenditen in % auf 1 Kommastelle gerundetQuelle: Fama/ French 1992, Tabelle I

Beta

Size


Fama-French Controversy I

• The Fama-French observation: if we control for size(capitalization), the positive relationship between beta and returndisappears.

• There is a joint B/M and size effect upon returns.• What measures the B/M ratio?• What measures size?


Fama-French Controversy II

• Misinterpretation of the results, time period: Black 1993• Inefficiency of market portfolio: Roll/Ross 1994 JF• Multifactor model: Fama/French 1996 JF• Outliers and trimmed regressions: Knez/Ready 1997 JF• The “alpha” factor: Ferson/ Sarkissian/ Simin 1999 JFM


Asset Pricing – New methodologies

• Multivariate tests: Gibbons 1982, Shanken 1985, etc.• APT-tests: Roll/Ross 1980, Dybvig/ Roll 1985, Roll/ Ross 1984,

Grinblatt/T itman 1985, …• Volatility and rational bounds: Shiller 1980, LeRoy/Porter, Marsh/

Merton, Cochrane, …• Time-series / Euler tests: Hansen/ Singleton 1982, 1983, …• Conditional CAPM: Ferson/ Harvey 1991 etc., Harvey 1991• Behavioral and experimental models: DeBondt/Thaler …


Performance Measurement: Jensen`s Alpha

R � 5%

R M � 10%

� M � 1�i

M

Ex-post Security Market Line

� �15.� � 0 5.

R

A

B

C

D

Epositive Performance

negative Performance


Other risk-adjusted performance measures

• The Sharpe ratio:Excess return in relation to total volatility

• The Treynor ratio:Excess return in relation to beta

• The appraisal ratio (Black/ Treynor):Alpha divided by specific risk


The Alpha in relation to Sharpe Ratios

A positive alpha requires

� �RRRR M

PortfolioofBeta

M

iiMi ��

��

�

��

which can be written as

��

BenchmarkofoSharpeRati

M

MiM

PortfolioofoSharpeRati

i

i RRRR�

��

��

�

The diversification effect is the major difference between Sharpe ratio comparisons and positive Alphas


The logic of the Appraisal ratio

• Performance measurement assumes active strategies, i.e. exante deviations from the benchmark.

• Therefore, alpha must be related to the active risk - the specificrisk or tracking error - of the portfolio.

• Equivalently, we can judge the manager-specific, i.e. non-marketperformance - that is �+��- by its Sharpe Ratio. But that SharpeRatio is the Appraisal Ratio!

• Problem: With a portfolio arbitrarily close to the benchmark, i.e.by minimizing the tracking error, the ratio can be inflated toinfinity - but this could contradict the portfolio strategy.


Performance Measures and T-Stats

• Sharpe Ratio is proportional to a t-stat whether the strategy‘sexcess return is different from zero. Without knowledge of therelevant benchmark, this is a sensible hurdle each strategyshould master. Proportionality factor is . Both measures leadto identical rankings (using the same number of observations, T)!

• Appraisal Ratio is proportional to t-stat whether alpha is differentfrom zero. Proportionality factor from OLS isBoth measures lead to identical rankings (using the sameobservations)!

� � ��

22

MMM RRRT

T


Performance – Early Tests

• The Friend/ Brown/ Herman/ Vickers 1962 SEC study: 152 funds1953-58, negative risk-adjusted performance of 20 bp, but costsof active management are approx. 100 bp, i.e. no overallinefficiency of the industry! No relationship between turnover andexpenses.


Performance – Classic tests

• Treynor/ Mazuy 1966 – negative performance, funds wasteresources

• Sharpe 1966 - negative performance, funds waste resources

• Jensen 1968 - negative performance, funds waste resources


Performance – Other Tests

• Friend/ Blume/ Crockett 1970 – positive alpha, contradict theclassic studies

• McDonald 1974 – small success in selectivity and timing• Carlson 1977 – contradicts Sharpe and Jensen• Mains 1977 – slightly positive alpha on average• Kon/ Jen 1979 - …evidence clearly inconsistent with Jensen• Shawky 1982 – zero performance, after costs• Ippolito 1989 – renowned for focus on trading, information and

management costs and results which contradict Sharpe andJensen. Alas, the results are biased due to data error


Performance – Econometric issues

• Timing/ tactical asset allocation – implies a nonlinear relationshipbetween beta and returns: Henriksson/Merton 1981, Veit/Cheney 1982, Kon 1983, Henriksson 1984, …

• Changing beta – missinterpretation of alphas: Grinblatt/Titman …• Data mining and survivorship: Brown/ Goetzman/ Ibbotson/ Ross

1992, Brown/Goetzman 1995, Blake/ Elton/ Gruber 1996• Sensitivity of results relative to benchmarks: Lehmann/Modest

1987• Using funds holdings data: Grinblatt/ Titman 1984• Factor models vs. equilibrium models, conditional and

unconditional


Performance – Practical issues

• Volatility is a bad risk measure if options are in the portfolio(hedge funds, structured products)

• Many funds are illiquid (hedge f., private eq., bond f.)• Sharpe ratio is used as marketing tool – but it is of limited value

to evaluate individual assets/asset classes in a portfolio context• Check the statistical significance of alphas, and translate it to a

time-horizon measure• Low correlation is used to promote diversification – but it

understates effective portfolio downside risk• Is performance persistent?• Survivorship bias


Tactical Asset Allocation: Bias in alpha


The misuse of the Sharpe Ratio

• The Sharpe Ratio (proposed by William Sharpe in 1966) is defined as theexpected or realized excess return divided by the standard deviation of theasset

• Correlation coefficients and diversification effects are not reflected in theSharpe Ratio

• The Sharpe Ratio is relevant only if an asset (class) is held individually or incombination with the riskless asset

• However, maximizing the Sharpe Ratio is reasonable for entire portfolios – thisyields the Tangency portfolio

• The size of the Sharpe Ratio for an individual asset (asset class) does not telltoo much about the optimal weight in a diversified portfolio


Numerical Exampleon the misuse of the Sharpe Ratio

Struktur des Tangentialportfolio CH-Aktien CH-Bonds INT-Aktien INT-Bonds

20.46% 24.80% 19.51% 35.22% Zugrundeliegende Annahmen: siehe Tabelle 1

Erwartete Rendite

Volatilität Sharpe Ratio

Korrelationskoeffizient

CH Aktien

CH Bonds

INT Aktien

INT Bonds

CH-Aktien 9% 18% 0.2778 1 CH-Bonds 5% 5% 0.2000 0.3 1 INT-Aktien 11% 22% 0.3182 0.5 0.1 1INT-Bonds 7% 12% 0.2500 0 0.4 0.2 1Tangentialp. 7.7% 9.1% 0.407


D

T

A

B C

Volatilität

F F

F’ E

T’

E

CML


Persistence

1537677

7652251982-84losers

7725521982-84winners

1985-87losers

1985-87winners

153 funds

Brown/Goetzman/Ibbotson/Ross (1992), Table 1Winner/loser: defined relative to median


Tests on persistence

• Cross product ratio: (52x52)/(25x25) = 4.24• Under the null, it should be 1.00, t-test clearly rejects

• Chi-square test (expected against actual values squared): Teststatistic is 18.35

• Under the null, it should be zero, �2-test clearly rejects

• And finally: a simple regression of subsequent alphas


Survivorship Bias

The Brown/ Goetzman/ Ibbotson/ Ross (1992) simulation:• Returns are generated by a market model – allowing for

dispersion in betas and unsystematic risk• Four-year returns are simulated over two subsequent time

periods• In each of the four years, the managers in the lowest percentile

indicated by the cutoff value are expluded from the sample• The experiment is repeated 20‘000 times.• Winner/loser: defined relative to median


The results of the simulation

0.80%4.681.927.1320%

0.61%3.361.373.2810%

0.44%2.051.161.645%

0.00%0.0041.011.04No cutoff

Avg. Ann.excessreturn

Avg. Crosssectional t-test

Avg. Crossproductratio

Avg. Chisquarestatistic

CutoffLevel

Brown/ Goetzman/ Ibbotson/ Ross (1992)


Survivorship Bias: Empirical estimates

The Elton/ Gruber/ Blake (1996) paper• Analyze 361 funds that exist in 1976• 106 merged, 216 survived, 39 restriced to public• 207 funds had more than 15 mio assets u.m.• Question 1: How to define „survival“: „not merge“ vs. „not merge

and keeping the same investment policy throughout the sample“• Question 2: How to calculate the performance of non-surviving

funds. Traditional approach vs. „follow the money“• Time period investigated: 1977-1993, Wiesenberger database• Three index model: Market, small stocks, bond yield


Survivorship Bias: Results

0.71% p.a.... and no change ininvestment policy

0.77% p.a.Did not mergeFollow the money**

0.73% p.a.... and no change ininvestment policy

0.91% p.a.Did not mergeTraditionalapproach*

Survivorship BiasQuestion 1Question 2

*Returns of non-surviving funds calculated up to and including the month of „death“.

**Assuming investing the money equally in all existing fundsElton/ Gruber/ Blake (1996)


Active versus passive investing

Aktive Underperformance: 1.2%


Active versus passive returns.


Expense ratios - institutional.

Ca. 0.8% Differenz


Expense ratios - private.

Ca. 1.2% Differenz


Turnover.


The impact of costs and taxes.

nominal Return1950-1999

Final Value1999

Stock Market 13.3% 514 $

Mutual Fundsincl. costs

11.1% 193 $

Mutual Fundsincl. costs/taxes

8.7% 65 $

Index Fundsincl. costs

13.1% 471 $

Index Fundsincl. costs/tax

11.7% 276 $

Figures from various Vanguard Sources, John Bogle Start 1950 mit 1 USD


Active vs. passive - a matter of style and timing

Zeitperiode: 1980-2000


Gute Zeiten für passives Mgmt...

• In 47% der Quartale der letzten 20 Jahre weisen die passivenManager eine Outperformance auf.

• Die Durchschnittsrendite des Marktes betrug in diesen Quartalen7%.

• In diesen Quartalen weisen LargeCaps eine Überperformancevon deutlich über 2% auf.

• ... und Growth wies gegenüber Value eine Überperformance vonrund 2% auf.


... und schlechte Zeiten.

• In 53% der Quartale der letzten 20 Jahre weisen die aktivenManager eine Outperformance auf.

• Die Durchschnittsrendite des S&P500 betrug lediglich 2% p.Q.• In diesen Quartalen weisen SmallCaps eine Überperformance

von 1.2% auf.• ... und Value wies gegenüber Growth eine Überperformance von

1.1% auf.


Mögliche Erklärung.

• Gibt es eine Erklärung für die zyklische Eigenschaft des Erfolgsindexierter Strategien?

• Kap.-gewichtete Indizes sind konstruktionsgemäss starkgewichtet in hochkapitalisierten Werten und Wachstumswerten,d.h. Indexierung bedeutet eine Selektivität gegen SmallCaps undValue Stocks

• ... die beide gerade in schlechten Märkten relativ gut rentieren.


Forts.

• Doch das typische aktive Portfolio vieler Investoren ist ehergleichgewichtet, d.h. besteht im Vergleich zum Index aus einerÜbergewichtung in Small Caps und Value Stocks.

• Deshalb schneiden aktive Strategien in steigenden Märktenschlechter ab als passive.

• Doch dies ist eine Hypothese.... Wie sieht die internationaleEvidenz dazu aus?

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